Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salpreimagtge | Structured version Visualization version GIF version |
Description: If all the preimages of left-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of left-closed, unbounded above intervals, belong to the sigma-algebra. (iii) implies (iv) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
salpreimagtge.x | ⊢ Ⅎ𝑥𝜑 |
salpreimagtge.a | ⊢ Ⅎ𝑎𝜑 |
salpreimagtge.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
salpreimagtge.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
salpreimagtge.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) |
salpreimagtge.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
salpreimagtge | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salpreimagtge.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | salpreimagtge.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
3 | salpreimagtge.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 1, 2, 3 | preimageiingt 40930 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} = ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵}) |
5 | salpreimagtge.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
6 | nnct 12780 | . . . 4 ⊢ ℕ ≼ ω | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≼ ω) |
8 | nnn0 39595 | . . . 4 ⊢ ℕ ≠ ∅ | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≠ ∅) |
10 | 3 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ) |
11 | nnrecre 11057 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ) | |
12 | 11 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ) |
13 | 10, 12 | resubcld 10458 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 − (1 / 𝑛)) ∈ ℝ) |
14 | salpreimagtge.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
15 | nfv 1843 | . . . . . . 7 ⊢ Ⅎ𝑎(𝐶 − (1 / 𝑛)) ∈ ℝ | |
16 | 14, 15 | nfan 1828 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) |
17 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆 | |
18 | 16, 17 | nfim 1825 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
19 | ovex 6678 | . . . . 5 ⊢ (𝐶 − (1 / 𝑛)) ∈ V | |
20 | eleq1 2689 | . . . . . . 7 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → (𝑎 ∈ ℝ ↔ (𝐶 − (1 / 𝑛)) ∈ ℝ)) | |
21 | 20 | anbi2d 740 | . . . . . 6 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ))) |
22 | breq1 4656 | . . . . . . . 8 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → (𝑎 < 𝐵 ↔ (𝐶 − (1 / 𝑛)) < 𝐵)) | |
23 | 22 | rabbidv 3189 | . . . . . . 7 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} = {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵}) |
24 | 23 | eleq1d 2686 | . . . . . 6 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → ({𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆)) |
25 | 21, 24 | imbi12d 334 | . . . . 5 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) ↔ ((𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆))) |
26 | salpreimagtge.p | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) | |
27 | 18, 19, 25, 26 | vtoclf 3258 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
28 | 13, 27 | syldan 487 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
29 | 5, 7, 9, 28 | saliincl 40545 | . 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
30 | 4, 29 | eqeltrd 2701 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 ≠ wne 2794 {crab 2916 ∅c0 3915 ∩ ciin 4521 class class class wbr 4653 (class class class)co 6650 ωcom 7065 ≼ cdom 7953 ℝcr 9935 1c1 9937 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 SAlgcsalg 40528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fl 12593 df-salg 40529 |
This theorem is referenced by: salpreimalelt 40938 salpreimagtlt 40939 issmfge 40978 |
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